Hydrodynamical Equation for Electron Swarms

We study the long time behavior of light particles, e.g. an electron swarm in which Coulomb interactions are unimportant, subjected to an external field and elastic collisions with an inert neutral gas. The time evolution of the velocity and position distribution function is described by a linear Boltzmann equation (LBE). The small ratio of electron to neutral masses, $\epsilon$, makes the energy transfer between them very inefficient. We show that under suitable scalings the LBE reduces, in the limit $\epsilon \to 0$, to a formally exact equation for the speed (energy) and position distribution of the electrons which contains mixed spatial and speed derivatives. When the system is spatially homogeneous this equation reduces to and thus justifies, for $\epsilon$ small enough, the commonly used ``two-term'' approximation.